Many modern communication systems are carrier-modulated systems that transmit information bearing signals through some specified bands. Due to the mismatch between the local oscillators of the transmitter and the receiver, a carrier frequency offset is seen at the receiver. In general, a large frequency offset causes rapid phase rotation, which could be far beyond the tracking capability of the receiver functions, such as the carrier phase tracking loop and the channel estimator. It is thus desired to have an initial frequency offset estimation to reduce the amount of frequency offset seen by the remaining part of the receiver. Some current frequency offset estimation algorithms are based on the ideal channel condition of additive white Gaussian noise (AWGN) and in some applications, a communications channel experiences severe inter-symbol interference (ISI). One example is the multi-path fading channel for wireless communications where ISI leads to extra noise terms and degrades the performance of prior art frequency offset estimations.
Several prior art frequency offset estimation schemes used in the ISI channels are briefly described herein. A “Fourier Coefficients after Non-linear Operation” algorithm performs the following steps:                Do some non-linear operation, such as differential detection,        Calculate the Fourier coefficients for some fundamental frequencies at the output of the non-linear operation,        Perform another operation on these coefficients.        Estimate the frequency offset from its phase.This algorithm does not require knowledge of received data symbols and timing, but it makes some approximations based on the assumption of the channel response. Another prior art scheme, “Jointly Channel and Frequency Offset Estimation”, utilizes a maximal-likelihood (ML) type of jointly channel and frequency offset estimator in which the algorithm has to search the maximal point of the non-linear ML equation after certain matrix operations are performed. This algorithm requires the knowledge of received data symbol and timing. The “Phase Rotation of Channel Estimation” method estimates frequency offset by neglecting the effect of frequency offset within a short time, then the frequency offset is estimated by the phase rotation of the channel coefficients. This method splits the channel and frequency offset estimations and requires knowledge of received data symbols and timing. The “Maximum Stated-based Accumulation (MSA)” method utilizes a modified version of the non-linear ML equation which is obtained by collecting the terms corresponding to the same state to form a whole while finding the maximal point of the modified non-linear equation. Knowledge about the received data symbols and timing is required.        